Congruent Circles Theorem

The biconditional statement will be proved separately.

If Two Circles Are Congruent, Then They Have the Same Radius

Consider two congruent circles $\odot A$ and $\odot B$ and a point on each one.

Because $\odot A\cong \odot B,$ the distance from the center to a point on the circle is the same for both circles. Therefore, $\overline{AC}\cong \overline{BD},$ which implies that both circles have the same radius. That is, $r_1=r_2.$

It has been proved that if two circles are congruent, then they have the same radius.

If Two Circles Have the Same Radius, Then They Are Congruent

Consider now two circles with the same radius.

Next, $\odot B$ can be translated so that point $B$ is mapped onto point $A.$ The image of $\odot B$ is $\odot B^{\prime },$ which is a circle centered at $A.$ Since the circles have the same radius, this translation maps $\odot B$ onto $\odot A.$ Because a rigid motion maps one circle onto the other, it is concluded that both circles are congruent.